The Triangular Hermite Finite Element Complementing the Bogner-Fox-Schmit Rectangle

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1 Applid Mtmtics Pblisd Onlin Dcmbr 3 (ttp://wwwscirporg/jornl/m) ttp://dxdoiorg/36/m3a6 T Tringlr Hrmit Finit Elmnt Complmnting t Bognr-Fox-Scmit Rctngl Lidi Gilv Vldimir Sydrov Boris Dobronts 3 Institt of Compttionl Modling Sibrin Brnc of Rssin Acdmy of Scincs Krsnoyrs Rssi Bing Univrsity Bijing Cin 3 Sibrin Fdrl Univrsity Krsnoyrs Rssi Emil: sidrov@milr Rcivd Octobr 3; rvisd Novmbr 3; ccptd Novmbr 7 3 Copyrigt 3 Lidi Gilv t l Tis is n opn ccss rticl distribtd ndr t Crtiv Commons Attribtion Licns wic prmits nrstrictd s distribtion nd rprodction in ny mdim providd t originl wor is proprly citd In ccordnc of t Crtiv Commons Attribtion Licns ll Copyrigts 3 r rsrvd for SCIRP nd t ownr of t intllctl proprty Lidi Gilv t l All Copyrigt 3 r grdd by lw nd by SCIRP s grdin ABSTRACT T Bognr-Fox-Scmit rctnglr lmnt is on of t simplst lmnts tt provid continos diffrntibility of n pproximt soltion in t frmwor of t finit lmnt mtod Howvr it cn b pplid only on simpl domin composd of rctngls or prlllogrms wos sids r prlll to two diffrnt strigt lins W propos nw tringlr Hrmit lmnt wit 3 dgrs of frdom It is sd in combintion wit t Bognr-Fox-Scmit lmnt nr t bondry of n rbitrry polygonl domin nd provids continos diffrntibility of n pproximt soltion in t wol domin p to t bondry Kywords: Continosly Diffrntibl Finit Elmnts; Bognr-Fox-Scmit Rctngl; Tringlr Hrmit Elmnt Introdction T finit lmnts wit intr-lmntl continos diffrntibility r mor complictd tn tos providing only continity Sc two-dimnsionl lmnts r mostly dvlopd for tringls: t Argyris tringl [] t Bll rdcd tringl [] t fmily of Morgn-Scott tringls [3] t Hsi-Clog-Tocr mcrotringl [] t rdcd Hsi-Clog-Tocr mcrotringl [5] t fmily of Dogls-Dpont-Prcll-Scott tringls [6] nd t Powll-Sbin mcrotringls [7] T Frijs d Vb-Sndr qdriltrl [8] nd its rdcd vrsion [9] r lso composd of tringls As for singl noncomposit rctngls t Bognr-Fox-Scmit (BFS) lmnt [] is t most poplr nd simplst on in t fmily of lmnts discribd by Zng [] All ts lmnts r widly sd in t conforming finit lmnt mtod for t birmonic qtion nd otr qtions of t fort ordr (s for xmpl [-8] nd rfrncs trin) long wit mixd sttmnts of problms nd nonconforming pproc [7] A dirct ppliction of t BFS-lmnt is rstrictd to t cs of simpl domin tt is composd of rctngls or prlllogrms wos sids r prlll to two diffrnt strigt lins Tis condition fils vn in t cs of simpl polygonl domin wr t intrsction of t bondry wit rctngls rslts in tringlr clls (cf Figr ) Of cors on cn constrct t spcil tringltion comptibl wit t bondry s som isoprmtric img [9] of domin composd of rctngls wit sids prlll to t xs Tis wy rqirs solving som dditionl bondry vl problms for t constrction of sc mpping tt is smoot ovr t wol domin In tis ppr w sggst to s t BFS-lmnts in t dirct wy (witot n isoprmtric mpping) for copl wit t proposd tringlr Hrmit lmnts wit 3 dgrs of frdom Ts tringlr lmnts spplmnt BFS-lmnts in t following sns Ty r sd only nr t bondry of polygonl domin nd provid intr-lmnt continos diffrntibility btwn finit lmnts of ts two typs Ts d to t joint s of ts lmnts on polygonl domin t pproximt soltion of finit lmnt mtod blongs to t clss C of fnctions wic r continosly diffrntibl on t closr of t domin Opn Accss

2 L GILEVA ET AL 5 Figr A sbdivision of domin Tringltion of Domin nd t Bognr-Fox-Scmit Elmnt Lt R b convx polygon wit bondry Assm tt w cn constrct tringltion of sbdividing it into closd rctnglr nd tringlr clls so tt t most of tm r rctngls nd only prt of t clls djcnt to t bondry my b rctnglr tringls Bsids ny two clls of my not v common intrior point nd ny two tringlr clls my not v common sid In ddition t nion of ll t clls coincids wit A simpl xmpl of t tringltion is sown in Figr W lso ssm tt ll sids of t rctnglr clls nd t ctti of t tringlr clls r prlll to t xs Dnot by t mximl dimtr of ll mss On rctnglr clls w s t Bognr-Fox- Scmit lmnt [] It is dfind by t tripl P wr is t rctngl wit vrtics 3 nd nd wit sid lngts nd Bsids P Q3 is t spc of bicbic polynomils on ; nd is t st of linr fnctionls (dgrs of frdom or nodl prmtrs) of t form [] p p p p p p p p i i i i i3 i i i p P i () T dimnsion of P Q3 (t nmbr of t cofficints of polynomil) is ql to 6 nd coincids wit t nmbr of t dgrs of frdom For tis lmnt w v t Lgrng bsis in P tt consists of t fnctions i jx i j stisfying t condition i j l i j l for i j l () wr is t Kroncr dlt Ts fnctions cn b writtn in t xplicit form wit t lp of t on-dimnsionl splins ˆ 3 3 nd ˆ 3 x x x x x x x Indd t dirct clcltions sow tt t bsis s t form ˆ ˆ x y ˆ x ˆ y ˆ x ˆ y 3 ˆ x ˆ y ˆ x ˆ y ˆ x ˆ y ˆ x ˆ y 3 ˆ x ˆ y ˆ x ˆ y 3 ˆ x ˆ y 3 ˆ x ˆ y 33 ˆ x ˆ y 3 ˆ x ˆ y ˆ x ˆ y ˆ x ˆ y 3 ˆ x ˆ y 3 T Rfrnc Tringlr Hrmit Elmnt First w constrct t rfrnc tringlr lmnt P ˆ ˆ ˆ wit t spcifid proprtis W considr t rigt tringl wic s nods ˆ i i wit t coordints ( ) ( ) (5 5) nd ( ) rspctivly (Figr ) W dfin t spc P of fnctions nd t st of dgrs of frdom s follows: P spn xˆ xˆ xˆ xˆ xˆ xˆ xˆ xˆ 3 3 ˆ xˆ xˆ xx ˆ ˆ xˆ xˆ xˆ xˆ xx ˆ ˆ 3 3 p p i pˆ pˆˆ ˆ pˆ pˆˆi p pi i ppˆ ˆ ˆ ˆ ˆ ˆ i i ˆ i i i3 ˆ ˆ ˆ ˆ ˆ i (3) Opn Accss

3 5 L GILEVA ET AL ˆx 3 Figr T rfrnc tringlr lmnt Obsrv tt t c of t nods ˆ ˆ nd tr r dgrs of frdom nd t t nod 3 tr is only on dgr Bsids t dgrs of frdom for t nods ˆ ˆ nd coincid wit tos for t nods of t BFS-lmnt () Lmm T tripl P ˆ ˆ ˆ is finit lmnt Pro of T dimnsion of t spc P coincids wit t nmbr of lmnts of t st ˆ Hnc to prov nisolvnc of t copl Pˆ ˆ it is sfficint to constrct t Lgrng bsis { ˆ ˆ ˆ i j x x Pˆ wr j for i = nd j = for i 3 } on stisfying t condition [] ˆ ˆ () i j l i j l T dirct clcltions sow tt t Lgrng bsis s t following form: 3 3 ˆ 3xˆ xˆ 3xˆ xˆ 3 3 ˆ xˆ xˆ xˆ 3xˆ xˆ xˆ xˆ xˆ xˆ 3 3 ˆ ˆ ˆ ˆ ˆ 3 xˆ 3xˆ xˆ xˆ xˆ x x x x 3 3 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ x x x x x x x ˆ ˆ ˆ ˆ x x x x x ˆ ˆ ˆ x x 3 xˆ xˆ 3 ˆ 3 3xx ˆ ˆ xx ˆ ˆ ˆ ˆ xx (5) 3 ˆ ˆ ˆ ˆ ˆ ˆ ˆ xx xx xx ˆ ˆ ˆ 3 6 xx 3 ˆ ˆ ˆ ˆ ˆ 3x x 8 x x 3 ˆ ˆ ˆ ˆ ˆ x x x x 3 ˆ ˆ ˆ ˆˆ ˆ ˆ 3 3xx xx x x 3 ˆ ˆˆ xx ˆˆ ˆ xx xxˆ Lt vˆ P ˆ b n rbitrry fnction Along t sid ˆ ˆ it is polynomil of dgr 3 in xˆ Togtr wit t drivtiv it is niqly dtrmind by t s ˆ v vl vˆ ˆ vˆ ˆ ˆ vˆ nd vˆˆ of nodl prmtrs In d dition t drivtiv ˆv lo ng ˆ ˆ is polynomil of dgr 3 in ˆx s wll nd is niqly dtrmind by t vls ˆ ˆ vˆ ˆ v ˆ v ˆx nd ˆ ˆ v of nodl prmtrs On t sid ˆ ˆ similr sttmnts r vlid witin t rplcmnt of ˆx by x ˆ Ts on t sids ˆ ˆ nd ˆ ˆ t vls of fnction of P nd its first-ordr t vls of nodl prmtrs of prtil drivtivs r niqly dtrmind by t t nods on t corrspondi ng sid On t sid ˆ ˆ similr pro prty dos not old Gnrlly sping t lmnt P ˆ ˆ ˆ is not of t clss C Tis follows from t fct tt t firstordr prtil drivtivs of t bsis fnctions rltd to t nod do not vnis on t sid ˆ ˆ Howvr frtr w ss m tt t sid of ny tringlr lmnt of bing t img of t ypotns ˆ ˆ is prt of t bo ndry nd cn not b common sid of two mss Hnc tis ftr s no inflnc on intrlmntl continity insid t domin To c c t intrpoltion proprtis w s t sl nottions for Sobolv spcs Hr H is t Hilbrt spc of fnctions Lbsg msrbl on qippd wit t innr prodct nd t finit norm v v v H d H For intgr nonngtiv H spc of fnctions H os w p to ordr inclsiv blong to H is t Hilbrt w drivtivs T norm in tis spc is dfind by t forml W lso s t sminorm s r s r srxx sr H s r srxx Lt û b n rbitrry fnction of H ˆ By Sobolv mbdding torm H ˆ is continosly mbddd into C ˆ [] nc ˆ C ˆ Ts w cn constrct its intrpolnt ˆ I P ˆ Dnot by mi i t nmbr of dgrs of frdom rltd to nod ˆ i W v m i ˆ ˆ ˆ ˆ ˆ ˆ I x x i j i j x x i j ˆ ˆ Torm Lt ˆ H ˆ Tn for ny intgr m w v t stimt ˆ ˆ c ˆ (6) I m ˆ ˆ wit constnt c indpndnt of û (nd ) Opn Accss

4 L GILEVA ET AL 53 Proof T mximl ordr of prtil drivtivs qls in t dfinition of t st ˆ As mntiond bov t spc H ˆ is mbddd into C ˆ Bsids from (3) it follows tt Pˆ P 3 ˆ wr P3 ˆ is t spc of polynomils of dgr no mor tn 3 Ts ll t ypotss of t Torm 35 in [] r flfilld wic implis t stimt (6) Combintion of Rctnglr nd Tringlr Elmnts Lt b n rbitrry tringlr lmnt (Figr 3) wit vrtics nd sid lngts T ffin mpping x x f ˆ ˆ x x tt mps t «rfrnc» lmnt into е s t form x xˆ x xˆ (7) W spcify t spc P of fnctions nd t st of dgrs of frdom s follows: 3 3 spn x x x x xx P x x 3 3 xx xx xx xx xx (8) p p i p p i i i i p p p p i i3 i i i T Lgrng bsis in P consists of t fnctions i jx wr j for i = nd j = for i = 3 bing t imgs of t bsis (5) ndr t mpping (7) nd stisfying t condition i j l i j l Ts t tripl P ffinly qivlnt to t «rfrnc» lmnt P ˆ ˆ (9) is finit lmnt tt is [] Now dnot t st of ll nods of t lmnts by nd nmbr tm from to s Wit c nod y w ssocit t nmbr m ql x to t nmbr of dgrs of frdom rltd to tis nod Obsrv tt m is oc crrd wn t nod y is t midpoint of t sid of tringlr lmnt bing prt of t bondry nd m for ll rmining nods of Tis is t globl nmbring of nods W lso intro- t locl nmbring T copl i is ssmd dc to b locl nmbr of t nod i of n lmnt To ny locl nmbr i tr corrsponds on nd only on globl nmbr nc w cn int fnction qi so tt qi In d- trodc dition for n lmnt w dnot t locl n log of t prm tr m by m i i m mi for qi At c nod y s w spcify m nmbrs v j j m Constrct t fnction v dfind on sc tt m i qi j i j v v j () i is niqly d- By t constrction t fnction find on c Pt v P if is rctngl P P if is tr ingl Tn v P for ny Lmm T fnction v dfind by t rltion () blongs to C Proof T BFS-rctngls r lmnts of t clss C i t fnction v nd its first-ordr drivtivs r continos on t sids common for two lmnts of tis typ [] Now lt b n rbitrry tringlr cll nd b rctnglr cll tt s common sid wit (Figr ) Bcs of constrction t vls of t nodl prmtrs of t fnctions v nd v on r ql In ddition t trcs of ts fnctions nd tir first-ordr drivt ivs wit rspct to x on r polynomils of dgr 3 in x nd r niqly dfind by t sts of nodl prm trs rltd to t s id Hnc t fnctions v nd v coincid on long wit tir first-ordr drivtivs x 3 β α Figr 3 A tringlr cll x Figr Two nigboring lmnts of diffrnt typs x Opn Accss

5 5 L GILEVA ET AL Corollry v H [] Ts w cn dfin t finit lmnt spc s fol- lows: V v C v P H Lt C Dfin its intrpolnt I V following wy: : m i I i j i j i j in t x x x x () Wit t lp of t Torm t following stimt cn b provd in t sl wy (s for instnc []) Torm Assm tt H And lt I V b its intrpolnt dfind by () Tn c m m I m Hr nd ltr constnts nd 5 Nmricl Exmpl r indpndnt of W illstrt proprtis of t proposd finit lmnts by t following xmpl Lt b rigt tringl wit nit ctti (s Figr 5 ) nd bondry Considr t problm c i f in wit t rigt-nd sid on cos sin f x y x y x y xy x y It s t xct soltion x y xysin x y Sbdivid t domin into lmntry sqrs (wit tringls djcnt to t potns) by drwing two fmilis of prlll strigt lins xi i i n nd yj j j n wit ms siz n To compr ccrcy wit dcrsing ms siz w constrct systm of linr lgbric qtion s of t finit lmnt mtod wit t BFS-lmnts on t lmntry sqrs nd t proposd lmnts on t tringls for n ndn 8 Sinc t xct soltion is nown t diffrnc btwn xct nd p- in t xplicit form proximt soltions cn b xprssd As rslt w v t following ccrcy Torticlly in t finit lmnt mtod w v t stimt [] c m m 3 I m Combining it wit t stimt in Torm w rriv t t following rror stimt for n pproximt soltion: c m m ( ) m Compring t lst two rslts in Tbl obsrv tt ty r clos to t symptotic vls nd 3 rspctivly 6 Smmry nd Frtr Implmnttions Ts t s of t proposd tringlr finit lmnts only nr bondry xtnds t fild of ppliction of t BFS-lmnts t lst for scond ordr qtions In its trn n pproximt soltion is of t clss H nbling on to clclt rsidl dirctly nd considrbly simplifis postriori ccrcy stimts for n pproximt soltion In principl to civ t sm ordr ccrcy inof t BFS-lmnts on cn s t Lgrng bicbic lmnts on rctngls nd t Lgrng l- std mnts of dgr tr on tringls Bt in tis cs gnrl dvntg of Hrmit finit lmnts in comprison wit Lgrng ons ms itslf vidnt in t nmbr of nnowns of discrt systm In prticlr Tbl sows t nmbr of dgrs of frdom for n pproximt soltion wic is ql to t nmbr of nnowns nd t nmbr of qtions in discrt systm of linr lgbric qtions for t xmpl from t bov sction Tbl Accrcy of n pproximt soltion log log Tbl T nmbrs of dgrs of frdom for n pproximt soltion Figr 5 Domin Ω wit initil tringltion Hrmit lmnts Lgrng lmnts Opn Accss

6 L GILEVA ET AL 55 3 b Figr 6 A nging nod b Obsrving considrbl gin in t nmbr of nnowns wit t frtr rfinmnt of tringltion t nmbr of nnowns (n d qtions) symptoticlly tnds to t rtio of :9 in fvor of Hrmit lmnts Now w notic n pprnt inconvninc nd sow wy to o vrcom it tt is lso sfl for nonconform ing grid rfinmnt T tringltion sown in Figr is constrctd by djsting clls of r ctnglr grid in t x - nd x -dirctions It imposs rst rictions on t rtio of stps for rctnglr grid insid domin W sow tt t proposd tringlr lmnts r sfficint to constrct conforming intrpolnt gnrlly sping on n ncoordintd grid witot rstriction on t rtio btwn stps Considring typicl cs sown in Fi gr 6 t no d b is so-clld nging nod At t lgo- lvl t cllng is solvd s follows For t ritmic dgrs of frdom t t nod b instd of t corrsponding vritionl qtions for n pproximt soltion w writ linr lgbric qlitis wic xprss t qntitis b b b nd b in trms of 6 dgrs of frdom of t nigboring rctngl Tis tric nsrs intrldiffrntibility of n intrpolnt b- mntl continos twn nd 3 Ts in tis cs on gin my implmnt proposd pir of lmnts in t frm of t conforming finit lmnt mtod wit stimt () REFERENCES [] J H Argyris I Frid nd D W Scrpf T TUBA Fmily of Plt Elmnts for t Mtrix Displcmnt Mtod Jornl of t Royl Aronticl Socity Vol 7 No pp 7-79 [] K Bll A Rfind Tringlr Plt Bnding Elmnt Intrntionl Jornl of Nmricl Mtods in Enginring Vol No 969 pp - ttp://dxdoiorg//nm68 [3] J Morgn nd L R Scott A Nodl Bsis for C Picwis Polynomils of Dgr n Mtmtics of Compttion Vol 9 No pp [] R W Clog nd J L Tocr Finit Elmnt Stiffnss Mtrics for Anlysis of Plts in Bnding Procdings of t Confrnc on Mtrix Mtods in Strctrl Mcnics Wrigt-Pttrson Air Forc Bs Oio Octobr 965 pp [5] P Prcll On Cbic nd Qrtic Clog-Tocr Finit Elmnts SI Jornl on Nmricl Anlysis Vol 3 No 976 pp -3 ttp://dxdoiorg/37/73 [6] J Dogls Jr T Dpont P Prcll nd R Scott A Fmily of C Finit Elmnts wit Optiml Approximtion Proprtis for Vrios Glrin Mtods for nd nd t Ordr Problms RA IRO Anlis Nmériq Vol 3 No pp 7-55 [7] M J D Powll nd M A Sbin Picwis Qdrtic Approximtions on Tringls ACM Trnsctions on Mtmticl Softwr Vol 3- No 977 pp ttp://dxdoiorg/5/ [8] B Frijs d Vb Bnding nd Strtcing of Plts Procdings of t Confrnc on Mtrix Mt- nd G S Holistr ods in Strctrl Mcnics Wrigt-Pttrson Air Forc Bs Oio Octobr 965 pp [9] B Frijs d Vb A Conforming Finit Elmnt for Plt Bnding In: J C Zinivicz Eds Str Anlysis Wily Nw Yor 965 pp 5-97 [] F K Bognr R L Fox nd L A Scmit T Gnr- Mss M- tion of Intrlmnt Comptibl Stiffnss nd trics by t Us of Intrpoltion Formls Procdings of t Confrnc on Mtrix Mtods in Strctrl Mcnics Wrigt-Pttrson Air Forc Bs Oio Octobr 965 pp 397- [] S Zng On t fll C -Q Finit Elmnt Spcs on Rctngls nd Cboids Advncs in Applid Mtmtics nd Mcnics Vol No 6 pp 7-7 [] P G Cirlt T Finit Elmnt Mtod for Elliptic Problms Nort-Hollnd Amstrdm 978 [3] Z C Li nd N Yn Nw Error Estimts of Bi-Cbic Hrmit Finit Elmnt Mtods for Birmonic Eqtions Jornl of Compttionl nd Applid Mtmtics Vol No pp 5-85 ttp://dxdoiorg/6/s377-7()9- [] S C Brnnr nd L R Scott T Mtmticl Tory of Finit Elmnt Mtods Springr-Vrlg Nw Yor 99 ttp://dxdoiorg/7/ [5] B M Irons A Conforming Qdrtic Tringlr Elmnt for Plt Bnding Intrntionl Jornl for Nmricl Mtods in Enginring Vol No 969 pp - [6] D L Logn A First Cors of Finit Elmnt Mtod SI Edition [7] J Blwndrd Plts nd FEM: Srpriss nd Pitflls Springr Nw Yor ttp://dxdoiorg/7/ [8] J Zo Convrgnc of V- nd F-cycl Mltigrid Mtods for Birmonic Problm Using t Hsi- Clog-Tocr Elmnt Nmricl Mtods for Prtil Diffrntil Eqtions Vol No 3 5 pp 5-7 ttp://dxdoiorg//nm8 Opn Accss

7 56 L GILEVA ET AL [9] J Ptr nd J F T Pittmn Isoprmtric Hrmit lmnts Intntionl Jornl for Nmricl Mtods in Enginring Vol 37 No 99 pp [] R A Adms nd J J F Fornir Sobolv Spcs Acdmic Prss Nw Yor 3 [] Z Cn nd H W Slctd Topics in Finit Elmnt Mtods Scinc Prss Bijing Opn Accss